The three levels of the theodolite were suitably delicate, the value of one division being 2·47″ (altitude), 4·92″ (transit), and 12·8″ (cross level). For these and every other level used, I adopted a distinctive system of numbering. Every level had a different number for the mean position of the bubble end, and the divisions were numbered uniformly in one direction, and not simply on each side of the mean. Thus the ranges were respectively from 5 to 15, 16 to 24, 28 to 32, 40 to 60, &c., on the levels called No. 10, No. 20, No. 30, No. 50, &c.; and when once a number was recorded (the mean of the two ends was always taken mentally), it showed which level was read, and in which direction, with any doubt, or further note.
Other adjuncts that I provided for this, and also for the other theodolites, were slit caps (see Figs. 6,7, 8, Pl. xv.). It is manifest that objects seen through a fine hole are always in equally good focus, no matter what may be the distance; hence, if an object-glass is limited to a small hole, it does not need focussing. But definition is commonly required in only one direction at once, either vertically or horizontally; hence a slit—which admits more light—will be as effective as a hole. When a line is quite invisible, by being out of focus, placing a slit cap over the object-glass, parallel with the line, will make it clear; and it will be well defined in proportion to the fineness of the slit. Each of the theodolites were therefore fitted with two movable slit caps, fine and wide, to cover the object-glasses. As focussing is always liable to introduce small errors, by shake of the tubes in each other, these slit caps were adopted to avoid the need of changing focus continually from near to distant objects; they also serve to bring near points in view, at only a foot or two from the glass. To be able to place the slit-cap on the end of the telescope, without shaking it, was essential. This I did by making the slit of thin steel spring; soldered to brass clutches, so as to grip the telescope by three points; provided also with a projecting tongue above, and another below it, whereby to bend it open for clipping it on (see Figs. 6, 7, 8, Pl. xv.). The smaller theodolites were also fitted with diagonal mirrors clipping on to the object-glass; these enabled the instruments to be very accurately centred without a plumb-line.
b. The 5-inch theodolite, by King, was an old one, and was obtained for rough work; but it had never been adjusted, so I had to take it in hand; and on finding its errors, after correction, to be even less than those of the 4-inch Troughton, I generally used it for all small work. I corrected it in the rectangularity of cones to the circles, of transit axis to the cones, and of cradle axis to transit axis; also in adjustment of verniers for run. The telescope was of long focussing-range when I got it, and I increased the range from infinite down to 5 1/2 feet focus, which made it very useful in near levelling, as in buildings; also I did away with the mere fit of sliding tubes for focussing; and made the inner tube run on four points, slightly punched up in the outer tube, and pressed in contact with them by a spring on the opposite side of it. The old level I replaced by a good one of Baker’s, running 41·5″ to ·1 inch. Microscopes of 1/4-inch equivalent focus were fitted to two arms, which were slipped together when required for use, and rode round on the compass-box; with these the average error of reading on the 1′ verniers was 7″.
The spider lines in this, and the next theodolite, were somewhat different to the usual pattern. When either a single vertical line, or a diagonal cross, is used, it blocks out any very small signal; and I have even heard of an engineer hunting in vain for his signal, because the line exactly hid it. To ensure greater accuracy, I therefore put in two parallel lines, crossed by one horizontal (needed for levelling); the lines being about 1/400 inch apart; if closer they may cling together if vibrated, and it is awkward to separate them while in the field. Thus the interval of the vertical lines was about 1′, and signals could be very accurately centred between them.*
c. The 4-inch theodolite by Troughton was not often used, except where lightness was important; I fitted it with two microscopes, similarly to the 5-inch; and its mean error of reading was about 8″ on the 1′ vernier.
Though neither of these were transit theodolites, yet in practice I used them as such for all accurate work. By reversing the telescope, end for end, and upside down, and turning the circle 180°, all the errors are compensated as in a transiting instrument; the only extra source of error is irregularity in the form of the rings, which can be tested by revolving the telescope in its cradle.
h. For ascertaining the angles of the Queen’s Chamber air channels I needed to measure as long a length of slope as possible, at about 8 feet inside a passage which was only 8 inches square. For this I pivoted an arm on the end of a long rod (see Fig. 9, Pl. xv.), and passed it into the passage in the dotted position at A; on reaching the slope it turned itself up to the angle by pressure, the main rod touching the passage roof. The arm carried an index, which touched a scale attached to the main rod. This scale was divided by actual trial, by applying a protractor to the limbs and marking the scale. To read it, a candle was carried on an arm, which shaded the direct light from the eye; and the scale was inspected by a short-focus telescope. Thus the readings were made without needing to withdraw the goniometer from the narrow channel, and hence the arm of it could be much longer than would be otherwise possible.
j. A large square, 35 and 45 inches in the sides, of sheet steel strips, 2 inches wide, and tinned together, I made for testing angles; it was not exactly adjusted to squareness, but its angles were very carefully fixed, by triangulating a system of fine punched dots on the face of it; and the edges were adjusted straight within about ·003 throughout their length. It could be used for the absolute value of slopes of about 51° 50′ and 26° 20′, by means of a rider level placed on one edge of it, and reading by means of a divided head screw at one end. To render the square stiff enough sideways, it was screwed down (with round projecting screw heads, not countersunk) to a frame of wooden bars, 2 × 1 inch in section. I generally found, however, that it was best to measure a slope by theodolite and offsets.
k. l. m. n. These stands were used for the theodolites. Generally the 10-inch theodolite could be placed directly on the rock, or on a stone; but when a stand was needed I used one about 30 inches high, that I made of 1 × 1 pine rod; the top was stouter and about 12 inches triangle, and the feet about 30 inches apart, connected by cross bars. Thus it was of the octahedral pattern, a triangular face at the top, another at the base, and six faces around; this being the only form absolutely free from racking. The screw feet of the theodolite rested on leaden trays on the top of the stand, which allowed free sliding for adjusting its centring. A similar octahedral stand about 16 high, was made of 1/2 × 1 inch pine, for the 5-inch theodolite; in order to stand it in chambers or on stones. The instrument was clamped on to the stand by a screw from beneath, passing through a plate under the triangular top of the stand, and screwing into the base plate of the theodolite, which rested upon the top of the stand. Thus it could be slid about